Given an array of non-negative integers, you are initially positioned at the first index of the array.

Each element in the array represents your maximum jump length at that position.

Your goal is to reach the last index in the minimum number of jumps.

**Example:**

Input:[2,3,1,1,4]Output:2Explanation:The minimum number of jumps to reach the last index is 2. Jump 1 step from index 0 to 1, then 3 steps to the last index.

Apply **memoization** to the previous solution. Your algorithm should run in **O(n^2)** time and use **O(n)** extra space.

The core challenge of this problem is to find the minimum number of jumps needed to reach the end of the array. Each element in the array tells you the maximum number of steps you can jump forward from that position. The goal is to determine the fewest jumps required to move from the start to the end of the array.

This problem is significant in scenarios where you need to minimize steps or actions to reach a goal, such as in game development or pathfinding algorithms.

To solve this problem, we can use a dynamic programming approach with memoization to store the results of subproblems and avoid redundant calculations.

A naive solution would involve trying every possible jump from each position and recursively calculating the minimum jumps needed to reach the end. This approach, however, would be highly inefficient with exponential time complexity.

We can optimize the naive solution by using memoization to store the minimum jumps needed from each position. This reduces the time complexity to O(n^2) and space complexity to O(n).

1. Initialize a memoization array to store the minimum jumps needed from each position.

2. Define a recursive function that calculates the minimum jumps from the current position to the end.

3. For each position, try all possible jumps and recursively calculate the minimum jumps needed from the new position.

4. Store the result in the memoization array to avoid redundant calculations.

```
public class JumpGameII {
// Memoization array to store the minimum jumps needed from each position
private int[] memo;
public int jump(int[] nums) {
int n = nums.length;
memo = new int[n];
// Initialize memo array with -1 indicating uncalculated positions
for (int i = 0; i < n; i++) {
memo[i] = -1;
}
// Start the recursive calculation from the first position
return minJumps(nums, 0);
}
private int minJumps(int[] nums, int position) {
int n = nums.length;
// If we have reached the last position, no more jumps are needed
if (position >= n - 1) {
return 0;
}
// If the result is already calculated, return it
if (memo[position] != -1) {
return memo[position];
}
int maxJump = nums[position];
int minSteps = Integer.MAX_VALUE;
// Try all possible jumps from the current position
for (int i = 1; i <= maxJump; i++) {
int nextPosition = position + i;
if (nextPosition < n) {
int steps = minJumps(nums, nextPosition);
if (steps != Integer.MAX_VALUE) {
minSteps = Math.min(minSteps, steps + 1);
}
}
}
// Store the result in the memo array
memo[position] = minSteps;
return minSteps;
}
public static void main(String[] args) {
JumpGameII solution = new JumpGameII();
int[] nums = {2, 3, 1, 1, 4};
System.out.println("Minimum jumps needed: " + solution.jump(nums)); // Output: 2
}
}
```

The time complexity of this approach is O(n^2) because for each position, we may need to check up to n possible jumps. The space complexity is O(n) due to the memoization array.

Consider edge cases such as:

- Array with a single element: [0]
- Array where the end is unreachable: [1, 0, 0, 0]
- Array with large jump values: [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]

Ensure the algorithm handles these cases correctly by testing with various inputs.

To test the solution comprehensively, use a variety of test cases, including simple, complex, and edge cases. You can use JUnit or any other testing framework to automate the testing process.

When approaching such problems, break down the problem into smaller subproblems and use dynamic programming to store intermediate results. Practice solving similar problems to improve your problem-solving skills.

Understanding and solving the Jump Game II problem helps in developing skills for dynamic programming and optimization techniques. Practice and explore further to master these concepts.